Fault-tolerant Locating-Dominating Sets on the Infinite King Grid
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Let G be a graph of a network system with vertices, V(G), representing physical locations and edges, E(G), representing informational connectivity. A locating-dominating (LD) set S is a subset of vertices representing detectors capable of sensing an "intruder" at precisely their location or somewhere in their open-neighborhood – an LD set must be capable of locating an intruder anywhere in the graph. We explore three types of fault-tolerant LD sets: redundant LD sets, which allow a detector to be removed, error-detecting LD sets, which allow at most one false negative, and error-correcting LD sets, which allow at most one error (false positive or negative). In particular, we determine lower and upper bounds for the minimum density of fault-tolerant locating-dominating sets in the infinite king grid; to prove the lower bounds, we introduce a new share-discharging strategy.
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